In the square root, we learn multiplication and division. Typically, in mathematics, we learn addition and subtraction first. But for square roots, multiplication and division are easier. We can just multiply the numbers in the root symbol.

However, there are rules for multiplication of square roots. And few people understand why it is okay to multiply numbers in the radical symbol by each other. You have to learn about square root multiplication and division, including the reasons why this is so.

Also, there is a method of calculation that is unique to square roots. That is prime factorization. Prime factorization makes the calculation easier and gives us the correct answer. In square root multiplication and division, we have to learn prime factorization at the same time.

In square root multiplication, we have to learn a new concept. We will explain how to multiply and divide square roots.

Table of Contents

## Methods of Square Root Multiplication and Division

As mentioned earlier in Square Root, we learn multiplication and division before addition and subtraction. Why do we learn multiplication in square roots first? It’s because multiplication between square roots is easy to do. Numbers in a radical sign can be multiplied.

For multiplication of the root sign, the formula is as follows.

So, for example, the calculation is as follows.

- $\sqrt{2}×\sqrt{3}=\sqrt{6}$
- $\sqrt{7}×\sqrt{11}=\sqrt{77}$

Also, division is the same as multiplication. This is because division can be converted to multiplication of fractions. Since square routes can be multiplied by each other, the numbers within a radical symbol can be divided in the same way.

Therefore, we calculate the division as follows.

- $\sqrt{21}÷\sqrt{3}=\sqrt{7}$
- $\sqrt{15}÷\sqrt{5}=\sqrt{3}$

In multiplication and division between square routes, we can do normal multiplication and division. The only difference is that radical symbols are there or not.

### Integers and Square Root Multiply and Divide Separately

The important thing is to separate integers and square roots, and multiply or divide them. Integers and square roots are completely different numbers.

In the square root it is described as $3\sqrt{2}$ and so on. This is a multiplication of 3 and $\sqrt{2}$, which has the same meaning as $3×\sqrt{2}$. But it is basic to omit the $×$, so it is written as $3\sqrt{2}$.

Because the numbers (integers) in front of the radical sign are considered separately, the integers and the square root have to be multiplied separately. In short, it is as follows.

An integer multiplies with an integer. A square root, on the other hand, multiplies with a square root. An integer does not multiply a number in the root sign. In multiplication and division, we need to calculate the integers and the square root separately.

So, for example, we have the following.

- $3×\sqrt{2}=3\sqrt{2}$
- $2×4\sqrt{2}=8\sqrt{2}$
- $2\sqrt{2}×3\sqrt{5}=6\sqrt{10}$

As you can see, the integer and the square root are calculated separately.

However, integers can be changed to square roots; we can convert integers to square roots by squaring them. For example, $2=\sqrt{2^2}=\sqrt{4}$. In the same way, $3=\sqrt{9}$. Also, $4=\sqrt{16}$.

We can’t multiply and divide an integer by a square root. So if we change the integer to a square root, we can multiply and divide the square roots by each other. For example, how can we solve the following problem?

- $\displaystyle\frac{\sqrt{12}}{2}$

$\sqrt{12}$ and 2 cannot be multiplied or divided. So, let’s use the root sign to represent 2. Since $2=\sqrt{4}$, we can calculate as follows.

- $\displaystyle\frac{\sqrt{12}}{2}=\displaystyle\frac{\sqrt{12}}{\sqrt{4}}=\sqrt{3}$

Integers and square roots cannot be multiplied. However, integers can be converted to square roots. So to be more precise, by converting an integer to a square root, you can multiply and divide an integer by a square root.

### Why Can We Multiply and Divide Between Square Routes

Why is it possible to multiply and divide route signs by each other? Few people understand the reason for this.

To understand this reason, we use a linear equation. Linear equations have the property that it doesn’t matter if we multiply both sides of the equation by the same number. For example, suppose we have the following balances.

If two are the same, equality holds no matter how many times both sides are multiplied.

To understand why the multiplication of the square root is valid, let’s review the properties of linear equations.

**-Prove Why Multiplication of the Square Root is Valid**

Next, let’s prove why it is possible to multiply and divide square roots by each other when calculating square roots. So let’s consider a situation where we multiply $\sqrt{a}$ and $\sqrt{b}$.

First, we square $\sqrt{a}×\sqrt{b}$. You must not ask, “Why do we square?” Anyway, squaring it. What happens when you square it? Because it is a multiplication, the meaning is the same with or without the parentheses. In multiplication, the parentheses can be removed at will.

For example, the following calculations have the same answer.

- $(2×3)×4=2×3×4$

This is called the associative property. Thus, the square of $\sqrt{a}×\sqrt{b}$ can be calculated as follows.

Thus, $\left(\sqrt{a}×\sqrt{b}\right)^2=a×b$.

Next, let’s convert this equation. First, we squared $\sqrt{a}×\sqrt{b}$. Then, return to the original state. Specifically, let’s add the root symbol to both sides.

Power is a type of multiplication. Also, integers can be converted to square roots by powers. In other words, it is a type of multiplication to make a square root from an integer. In an equation, we can multiply both sides of the equation by the same number. Let’s use this property to put a radical sign on both sides of the equation. Then we get the following result.

In this way, we have proved that $\sqrt{a}×\sqrt{b}=\sqrt{a×b}$. We can prove that we can multiply and divide square roots by using our previous knowledge of mathematics.

## Use Prime Numbers and Do Prime Factoring

Note that the number in the radical symbol has to be as small as possible. How can we do this? We can make the square root by squaring the integer. Similarly, if there is a square in the root, we can convert it to an integer and then move it out of the root symbol.

Not only can we make a square root from an integer, but we can also make an integer from part of the square root. For example, we have the following.

- $\sqrt{12}=\sqrt{2×2×3}=\sqrt{2^2×3}=2\sqrt{3}$

How can we efficiently find powers in the root symbol? The method is prime factorization. Prime factorization is to use prime numbers to perform division.

Numbers have prime numbers. A prime number is a number that is divisible only by 1 and its number. Prime numbers include, for example the following.

- Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19

For example, 7 has only 1 and 7 divisors. Likewise, the divisors for 11 are only 1 and 11. Dividing by any other number does not result in an integer. On the other hand, the divisors of 4 are 1, 2, and 4. Not only 1 and 4, but 2 is also a divisor. Therefore, 4 is not a prime number.

We can convert an integer (natural number) into a multiplication equation by dividing it by a prime number. Understand that the opposite of multiplication calculations is prime factorization.

When we think of prime factorization, the term is difficult to understand. However, the concept of prime factorization is simple if you understand it as a way to decompose numbers into multiplication expressions. Prime numbers cannot be further divisible. So, with prime numbers, we can convert them into multiplication.

### How to Do Prime Factorization? Steps to multiply

How to do prime factorization? As an example, let’s try to factor 300 into prime numbers.

The way to do it is to divide in order from the smallest prime number. The smallest prime number is 2. So, let’s divide 300 by 2. If we divide 300 by 2, we can divide it into 2 and 150. $300=2×150$.

This method allowed us to do prime factorization once. However, 150 is not a prime number. It can be further divided by a prime number. So, let’s divide it again by a prime number. Dividing by 2, we get the following.

For 300, divide twice by 2 to get 75. That is, $300=2×2×75$. However, 75 is not a prime number. So, let’s do more prime factorization.

75 is not divisible by 2. So, next we divide it by 3.

The result is $300=2×2×3×25$. In the same way, we can also proceed with prime factorization. 25 is not a prime number, and can be divided by 5. Therefore, we get the following

Dividing by the prime numbers, we finally found the following facts.

- $300=2×2×3×5×5$

Since 5 is a prime number, it cannot be divided any further. In prime factorization, stop the division when the prime number appears at the end. Then collect all the divided prime numbers and the last prime number. Since multiplying all the prime numbers in the circle in the figure above to get 300, the prime factorization is completed.

**-The Symbol for Prime Factorization**

When we actually do prime factorization, we have to calculate it efficiently and quickly. Therefore, we almost never use arrows to do prime factorization as shown above. Instead, we use the following symbol.

In its usage, write the integers (natural numbers) inside the symbol. Also, write a prime number to the left of the symbol. It looks like this.

In terms of usage, it’s the same as the previous arrow diagram. We decompose an integer by dividing it by a prime number. For example, a prime factorization of 300 would look like this.

If the prime numbers appear at the end, the prime factorization is finished. Then, by collecting all the prime numbers, we can make a multiplication equation. The meaning is the same as the arrow diagram. The difference is that by using this symbol, we can do prime factorization more efficiently.

### The Answer to Prime Factorization Is the Same in Any Order

By the way, prime factorization can be calculated in any order. Because the final answer will be the same. For example, the prime factorization of 300 will give the same answer even if the order of division is different, as shown below.

We have explained earlier that, in prime factorization, we should divide the prime numbers in the order of their smaller numbers. This is because the smaller the prime number, such as 2 or 3, the easier it is to divide, and the fewer mistakes are made. However, in some cases, it may be better to divide by a larger prime number to avoid miscalculation.

If the integer we want to divide is an even number, we can always divide it by 2. On the other hand, if it’s an odd number, it’s sometimes unclear if we can divide it by a smaller prime number.

For example, what prime number can 105 be divided by. 105 can be divided by 3. However, it is difficult to notice that it is divisible by 3. On the other hand, it’s easy to notice that 105 is divisible by 5.

The knowledge that we can do prime factorization in any order can prevent miscalculation in mathematics. By using prime factorization, we can return an integer to a multiplication equation. In this case, don’t worry about the order in which you divide by prime numbers.

### Take Out the Square Number Inside the Radical Symbol

Why do we need to learn prime factorization? It’s because we use prime factorization a lot in the calculation of the square root.

In the calculation of square root, there is a rule that the number in the root symbol must be as small as possible. In order to do so, we have to do prime factorization. Specifically, we have to take out the squared numbers in the root symbol.

As mentioned earlier, if there are squares in the radical symbol, we can put the numbers outside the root sign.

By using this formula, we can put the numbers outside in the root sign after doing prime factorization. For example, we have already calculated that prime factorization of 300 yields the following

- $300=2×2×3×5×5=2^2×3×5^2$

So how can $\sqrt{300}$ be calculated? Consider $\sqrt{300}$ as follows to get the number out of the radical sign.

The reason for learning prime factorization is that it is used in square root calculations. When you take the numbers in the root symbol out, everyone does prime factorization.

So why do we need to take the numbers in the radical symbol out? The reason is simple: it’s easier to understand the numbers. For example, the number $\sqrt{300}$ is difficult to answer instantly. On the other hand, $10\sqrt{3}$ is easy.

$\sqrt{3}≒1.73$. Since it is 10 times that, we know that $10\sqrt{3}$ is about 17.3. Mathematics is the study of understanding what numbers mean. We have to make the numbers easy to understand, so we have to get the numbers out of the root symbol.

### Do not Divide Other Than Prime Numbers

Note that in prime factorization, we should not use non-prime numbers for division. If we mix non-prime numbers, we can’t take out the numbers in the radical sign. For example, when we do prime factorization of 300, what happens if we divide it by 4?

4 is not a prime number because it can be divided by 2. However, if we divide by 4, many people make the following equation.

- $\sqrt{300}=\sqrt{3×4×5^2}$

As a result, we end up with an answer of $5\sqrt{12}$. This answer is wrong, because we can make the number in the route symbol even smaller.

For a non-prime number, such as 4, we can divide it by another number. As a result, we can’t change to a multiplication equation using the smallest number. Thus, the numbers in the root sign become larger and cause a calculation mistake. In prime factorization, be sure to divide by prime numbers.

## Learn How to Multiply and Divide Between Square Routes

In mathematics, it is often necessary to multiply and divide square roots by each other. In such cases, prime factorization makes it easier to get the numbers in the root symbol out.

For example, how can we calculate the following.

- $\sqrt{15}×\sqrt{21}$

Prime factorizing 15 is $3×5$. And if we prime factorize 21, it is $3×7$. Therefore, we can calculate as follows.

$\sqrt{15}×\sqrt{21}$

$=\sqrt{3×5}×\sqrt{3×7}$

$=\textcolor{red}{\sqrt{3}}×\sqrt{5}×\textcolor{red}{\sqrt{3}}×\sqrt{7}$

$=3×\sqrt{5×7}$

$=3\sqrt{35}$

When multiplying, prime factorization reveals that $\sqrt{15}$ and $\sqrt{21}$ contain $\sqrt{3}$ in the root sign, respectively; we can make 3^{2}, so we can get 3 out of the radical symbol. By prime factorizing, we can understand which numbers can be outside of the root sign.

Note that if we cannot create powers, we cannot get the numbers out of the radical sign. In the previous calculation, the inside of the root sign is calculated as $\sqrt{5×7}=\sqrt{35}$. This is because although a square exists for 3, we cannot create a square for 5 and 7.

### Perform a Prime Factorization of the Square Root First

By the way, in multiplication and division of a square route, the first thing we should do is to make the numbers in the radical symbol smaller. In both multiplication and division, the smaller the number, the fewer calculation mistakes occur. For this reason, it is best to do prime factorization first in square root calculations.

For example, how can the following problem reduce miscalculation?

- $\sqrt{28}×\sqrt{18}$

There are two ways to calculate this as follows.

Checking the two calculation methods, it is obvious that the calculation is easier to do with the prime factorization first. Multiplication of 28 and 18 is complicated. Also, doing the prime factorization of 504 is hard.

For this reason, you should always do prime factorization first, whether in multiplication and division. That way, you’ll make fewer mistakes in your calculations.

## Exercises: Square Root Multiplication and Division

**Q1.** Do the following calculations.

- $2\sqrt{3}×4\sqrt{21}$
- $\sqrt{15}×\sqrt{30}$
- $\sqrt{75}÷2\sqrt{3}$
- $(3+\sqrt{5})(3-\sqrt{5})$

**A1.** Answers.

**(a)**

$2\sqrt{3}×4\sqrt{21}$

$=2\sqrt{\textcolor{red}{3}}×4\sqrt{\textcolor{red}{3}×7}$

$=2×4×3×\sqrt{7}$

$=24\sqrt{7}$

**(b)**

$\sqrt{15}×\sqrt{30}$

$=\sqrt{\textcolor{red}{3}×\textcolor{blue}{5}}×\sqrt{2×\textcolor{red}{3}×\textcolor{blue}{5}}$

$=3×5×\sqrt{2}$

$=15\sqrt{2}$

**(c)**

If there is a division, be sure to correct it with the multiplication of fractions.

$\sqrt{75}÷2\sqrt{3}$

$=\sqrt{3×5×5}×\displaystyle\frac{1}{2\sqrt{3}}$

$=5\textcolor{red}{\sqrt{3}}×\displaystyle\frac{1}{2\textcolor{red}{\sqrt{3}}}$

$=\displaystyle\frac{5}{2}$

**(d)**

In the calculation of the square root, we may utilize the Factoring Formula. The formula is following.

- $(x+a)(x-a)=x^2-a^2$

So let’s calculate the square root by using the formula $(x+a)(x-a)=x^2-a^2$.

$(3+\sqrt{5})(3-\sqrt{5})$

$=3^2-(\sqrt{5})^2$

$=9-5$

$=4$

## Multiply the Square Root and Do Prime Numbers Calculation

Square roots calculation frequently involves multiplication and division. Square roots can multiply and divide each other. The method is the same as in normal multiplication. However, it is important to understand the rules of multiplying and dividing integers and square roots separately.

Note that we must always do prime factorization when calculating the square root. We have to convert an integer into a multiplication equation. By doing prime factorization, we can get the numbers in the root symbol out.

Also, by doing prime factorization first, we can make the square root calculation easier. There are fewer calculation mistakes when we do prime factorization first. Let’s understand how to reduce miscalculation, including how to do it.

When you multiply and divide square roots, we have explained not only how but also why the calculation is done in that way. By learning these, you will be able to perform mathematical calculations, including square roots.